Neither the first nor the second input give a value to me, but
In[41]:= (* x Rotation *)
\[Pi] Integrate[(-2 x + 1)^2, {x, -2, 0}]
Out[41]= (62 \[Pi])/3
In[42]:= (* y Rotation *)
\[Pi] Integrate[(-1/2 (y - 1))^2, {y, 0, 1}] + \[Pi] Integrate[(1/2 (y - 1))^2, {y, 1, 5}]
Out[42]= (65 \[Pi])/12
because in cylindrical co-ordinates the volume is
$\int_a^b\int_0^{2\pi}\int_0^{f(z)} r dr d\varphi dz = 2\pi \int_a^b \frac{1}{2} (f(z))^2 dz = \pi \int_a^b (f(z))^2dz$ where
$f(z)$ is understood to be non-negative, i.e. if
$f(z)$ has a zero one has to stop integration at it and continue with the next positive part of the boundary description becoming
$f(z)$ and so on.
